Average Error: 0.2 → 0.0
Time: 15.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[4 \cdot \left(b \cdot b\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
4 \cdot \left(b \cdot b\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)
double f(double a, double b) {
        double r7019893 = a;
        double r7019894 = r7019893 * r7019893;
        double r7019895 = b;
        double r7019896 = r7019895 * r7019895;
        double r7019897 = r7019894 + r7019896;
        double r7019898 = 2.0;
        double r7019899 = pow(r7019897, r7019898);
        double r7019900 = 4.0;
        double r7019901 = r7019900 * r7019896;
        double r7019902 = r7019899 + r7019901;
        double r7019903 = 1.0;
        double r7019904 = r7019902 - r7019903;
        return r7019904;
}
↓
double f(double a, double b) {
        double r7019905 = 4.0;
        double r7019906 = b;
        double r7019907 = r7019906 * r7019906;
        double r7019908 = r7019905 * r7019907;
        double r7019909 = -1.0;
        double r7019910 = a;
        double r7019911 = r7019910 * r7019910;
        double r7019912 = r7019911 + r7019907;
        double r7019913 = sqrt(r7019912);
        double r7019914 = pow(r7019913, r7019905);
        double r7019915 = r7019909 + r7019914;
        double r7019916 = r7019908 + r7019915;
        return r7019916;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\left(-1 + \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right) + \left(b \cdot b\right) \cdot 4}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

    \[\leadsto \left(-1 + \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \left(a \cdot a + b \cdot b\right)\right) + \left(b \cdot b\right) \cdot 4\]

  5. Applied associate-*l*0.1

    \[\leadsto \left(-1 + \color{blue}{\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(a \cdot a + b \cdot b\right)\right)}\right) + \left(b \cdot b\right) \cdot 4\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

    \[\leadsto \left(-1 + \sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}\right)\right) + \left(b \cdot b\right) \cdot 4\]

  8. Applied cube-unmult0.1

    \[\leadsto \left(-1 + \sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}}\right) + \left(b \cdot b\right) \cdot 4\]

  9. Applied pow10.1

    \[\leadsto \left(-1 + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1}} \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}\right) + \left(b \cdot b\right) \cdot 4\]

  10. Applied pow-prod-up0.0

    \[\leadsto \left(-1 + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\left(1 + 3\right)}}\right) + \left(b \cdot b\right) \cdot 4\]

  11. Simplified0.0

    \[\leadsto \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\color{blue}{4}}\right) + \left(b \cdot b\right) \cdot 4\]

  12. Final simplification0.0

    \[\leadsto 4 \cdot \left(b \cdot b\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))